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Properties

Label 2-338-13.5-c2-0-4
Degree $2$
Conductor $338$
Sign $0.697 - 0.716i$
Analytic cond. $9.20983$
Root an. cond. $3.03477$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 3.84·3-s − 2i·4-s + (−3.77 + 3.77i)5-s + (3.84 − 3.84i)6-s + (−7.25 − 7.25i)7-s + (2 + 2i)8-s + 5.80·9-s − 7.54i·10-s + (−7.40 − 7.40i)11-s + 7.69i·12-s + 14.5·14-s + (14.5 − 14.5i)15-s − 4·16-s − 4.88i·17-s + (−5.80 + 5.80i)18-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 1.28·3-s − 0.5i·4-s + (−0.754 + 0.754i)5-s + (0.641 − 0.641i)6-s + (−1.03 − 1.03i)7-s + (0.250 + 0.250i)8-s + 0.644·9-s − 0.754i·10-s + (−0.673 − 0.673i)11-s + 0.641i·12-s + 1.03·14-s + (0.967 − 0.967i)15-s − 0.250·16-s − 0.287i·17-s + (−0.322 + 0.322i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.697 - 0.716i$
Analytic conductor: \(9.20983\)
Root analytic conductor: \(3.03477\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1),\ 0.697 - 0.716i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.280395 + 0.118279i\)
\(L(\frac12)\) \(\approx\) \(0.280395 + 0.118279i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
13 \( 1 \)
good3 \( 1 + 3.84T + 9T^{2} \)
5 \( 1 + (3.77 - 3.77i)T - 25iT^{2} \)
7 \( 1 + (7.25 + 7.25i)T + 49iT^{2} \)
11 \( 1 + (7.40 + 7.40i)T + 121iT^{2} \)
17 \( 1 + 4.88iT - 289T^{2} \)
19 \( 1 + (18.6 - 18.6i)T - 361iT^{2} \)
23 \( 1 + 19.9iT - 529T^{2} \)
29 \( 1 - 14.3T + 841T^{2} \)
31 \( 1 + (19.0 - 19.0i)T - 961iT^{2} \)
37 \( 1 + (-42.9 - 42.9i)T + 1.36e3iT^{2} \)
41 \( 1 + (3.54 - 3.54i)T - 1.68e3iT^{2} \)
43 \( 1 - 11.9iT - 1.84e3T^{2} \)
47 \( 1 + (-7.59 - 7.59i)T + 2.20e3iT^{2} \)
53 \( 1 - 77.0T + 2.80e3T^{2} \)
59 \( 1 + (44.4 + 44.4i)T + 3.48e3iT^{2} \)
61 \( 1 + 56.2T + 3.72e3T^{2} \)
67 \( 1 + (-4.32 + 4.32i)T - 4.48e3iT^{2} \)
71 \( 1 + (-40.4 + 40.4i)T - 5.04e3iT^{2} \)
73 \( 1 + (12.7 + 12.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 7.98T + 6.24e3T^{2} \)
83 \( 1 + (-35.8 + 35.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (57.2 + 57.2i)T + 7.92e3iT^{2} \)
97 \( 1 + (-38.7 + 38.7i)T - 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.06930904820075471458524998819, −10.63046085914713538602938772195, −9.953506708031994650517324038375, −8.423310388586333715026756022296, −7.40508944725798534002878998251, −6.58014728622117955306393774176, −5.96715174904956560732823706228, −4.55060151144800266118931414659, −3.22100230225897114873029116215, −0.53548207358783768230381120645, 0.42413464395360098063713175503, 2.49584005566626208371210000692, 4.16388648253354861265056838083, 5.27555263517603360644157999479, 6.25014087271511344730230906897, 7.42158127388600240154389675818, 8.619148828203288789282966842145, 9.391846024462618135033455428004, 10.43965957554198032779636734580, 11.30176114683832253698559033822

Graph of the $Z$-function along the critical line